A Model for Protostellar Cluster Luminosities and the Impact on the CO-H$_2$ Conversion Factor

We construct a semi-analytic model to study the effect of far-ultraviolet (FUV) radiation on gas chemistry from embedded protostars. We use the Protostellar Luminosity Function (PLF) formalism of Offner & McKee (2011) to calculate the total, FUV, and ionizing cluster luminosity for various protostellar accretion histories and cluster sizes. We compare the model predictions with surveys of Gould Belt star-forming regions and find the Tapered Turbulent Core model matches best the mean luminosities and the spread in the data. We combine the cluster model with the photo-dissociation region astrochemistry code, {\sc 3d-pdr}, to compute the impact of the FUV luminosity from embedded protostars on the CO to H$_2$ conversion factor, $X_{\rm CO}$, as a function of cluster size, gas mass and star formation efficiency. We find that $X_{\rm CO}$ has a weak dependence on the FUV radiation from embedded sources for large clusters due to high cloud optical depths. In smaller and more efficient clusters the embedded FUV increases X$_{\rm CO}$ to levels consistent with the average Milky Way values. The internal physical and chemical structure of the cloud are significantly altered, and $X_{\rm CO}$ depends strongly on the protostellar cluster mass for small efficient clouds.